ABSTRACT:Let n be a positive integer and λ > 0 a real number. Let V n be a set of n points in the unit disk selected uniformly and independently at random. Define G(λ, n) to be the graph with vertex set V n , in which two vertices are adjacent if and only if their Euclidean distance is at most λ. We call this graph a unit disk random graph. Let λ = c √ ln n/n and let X be the number of isolated points in G(λ, n). We prove that almost always X ∼ n 1−c 2 when 0 ≤ c < 1. It is known that if λ = √ (ln n + φ(n))/n where φ(n) → ∞, then G(λ, n) is connected. By extending a method of Penrose, we show that under the same condition on λ, there exists a constant K such that the diameter of G(λ, n) is bounded above by K ·2/λ. Furthermore, with a new geometric construction, we show that when λ = c √ ln n/n and c > 2.26164 · · · , the diameter of G(λ, n) is bounded by (4 +o(1))/λ; and we modify this construction to yield a function c(δ) > 0 such that the diameter is at most 2(1+δ+o(1))/λ when c > c(δ).
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